58 Jean-Marie De Koninck

x N(x)

10 10

100 33

1000 213

x N(x)

104

1 538

105

11 872

106

95 428

x N(x)

107

806 095

108

6 954 793

109

61 574 510

• the smallest number n such that λ0(n) = λ0(n + 1) = . . . = λ0(n + 7) = 1,

where λ0 stands for the Liouville function; moreover, in this case, we also have

λ0(n + 8) = 1 (see the number 1 934 for the list of the smallest numbers at

which the λ0 function takes the value 1 at consecutive arguments).

215

• the second solution of σ2(n) = σ2(n + 2) (see the number 1 079).

216

• the only cube which can be written as the sum of the cubes of three consecutive

numbers: 216 = 63 = 33 + 43 + 53;

• the tenth 3-powerful number (counting 1 as a 3-powerful number); if nk stands

for the kth 3-powerful number, then n10 = 216, n100 = 52 488, n1

000

=

25 153 757, n10

000

= 16 720 797 973, n100

000

= 13 346 039 198 336 and

n1

000 000

= 11 721 060 349 748 875.

217

• the smallest number 1 for which the sum of its divisors is a fourth power:

σ(217) = 44; the sequence of numbers satisfying this property begins as follows:

217, 510, 642, 710, 742, 782, 795, 862, 935, . . . ; it is also the smallest number

n such that σ(n) is an eighth power: σ(217) = 28.

220

• the number which, when paired with the number 284, forms the smallest ami-

cable pair: two numbers are called amicable if the sum of the proper divisors of

one of them equals the other: here σ(220) − 220 = 284 and σ(284) − 284 = 220.

221 (=

102

+

112

=

52

+

142)

• the number used by Euler to show how to find its factors given that two distinct

representations of it as the sum of two

squares70

are known.

70Here is Euler’s method. Let n be an odd number which can be written as the sum of two squares

in two distinct ways:

n =

a2

+

b2

=

c2

+

d2,